I've noticed that most proofs of the density of $\mathbb Q$ in $\mathbb R$ use the Archimedean principle. For example see @Arturo Magidin's checked answer here. Density of irrationals
I'm puzzled by the fact that in the hyperreals, the standard rationals must be dense in the standard reals; yet the hyperreals are not Archimedean. This would seem to imply that the Archimedean principle is not a necessary assumption.
I can see that in the hyperreals, if $\epsilon$ is infinitesimal then there is no standard rational between $\epsilon$ and $2 \epsilon$; so that the standard rationals are not dense in the hyperreals.
But now I'm confused. Wouldn't the hyper-rationals be dense in the hyperreals? And wouldn't it still be a theorem that the standard rationals are dense in the standard reals?
Clearly the answer must be in some subtlety involving standard/nonstandard reals and the transfer principle. Can anyone shed any light? Is the Archimedean principle necessary to prove that the rationals are dense in the reals? Why aren't the hyperreals a counterexample?